Problem 63 - Chemistry The Science in Context

Question

Which of the following actions would produce the greater increase in the volume of a gas sample: (a) lowering the pressure from \(760 \mathrm{mmHg}\) to \(720 \mathrm{mmHg}\) at constant temperature or (b) raising the temperature from \(10^{\circ} \mathrm{C}\) to \(40^{\circ} \mathrm{C}\) at constant pressure?

Step-by-Step Solution

Answer

Short Answer: The greater increase in the volume of the gas sample can be determined by comparing the volume ratios for both cases. In case (a), after converting the pressures to Pascals and using the simplified Ideal Gas Law, the volume ratio (V2/V1) is calculated. In case (b), after converting the temperatures to Kelvin and using the simplified Ideal Gas Law, the volume ratio (V2/V1) is calculated. By comparing the volume ratios from both cases, the action with the greater volume ratio will produce the greater increase in the volume of the gas sample.

Step 1: Identify the givens for case (a) and case (b) and convert to SI units

For case (a), we are given: - Initial pressure, \(P_1 = 760 \mathrm{mmHg}\) - Final pressure, \(P_2 = 720 \mathrm{mmHg}\) Convert both pressures to Pascals (Pa) using the relation \(1 \mathrm{mmHg} = 133.322 \mathrm{Pa}\). For case (b), we are given: - Initial temperature, \(T_1 = 10^{\circ} \mathrm{C}\) - Final temperature, \(T_2 = 40^{\circ} \mathrm{C}\) Convert both temperatures to Kelvin (K) using the relation \(T (K) = T (°C) + 273.15.\)

Step 2: Apply the Ideal Gas Law for case (a) and find the ratio of final to initial volume

Since the temperature and the quantity of gas are constant, we can simplify the Ideal Gas Law for case (a) as follows: \(\frac{P_1}{P_2} = \frac{V_2}{V_1}\) Now, calculate the ratio \(\frac{V_2}{V_1}\) for case (a).

Step 3: Apply the Ideal Gas Law for case (b) and find the ratio of final to initial volume

Since the pressure and quantity of gas are constant for case (b), we can simplify the Ideal Gas Law as follows: \(\frac{T_1}{T_2} = \frac{V_2}{V_1}\) Now, calculate the ratio \(\frac{V_2}{V_1}\) for case (b).

Step 4: Compare the volume ratios for both cases and determine the greater increase

Compare the volume ratios calculated in Step 2 and Step 3. The action with the greater volume ratio will produce the greater increase in the volume of the gas sample.

Key Concepts

Volume Change

When we talk about the volume change of a gas, we are essentially interested in how the gas expands or contracts under certain conditions like same temperature or constant pressure. For example, in our exercise, two scenarios require analysis to understand which causes a greater change in volume. By analyzing volume change, we compare how a gas sample behaves when pressure is decreased versus when temperature is increased.

Here's how they affect volume:

• Decreasing the pressure of the gas while keeping the temperature constant will likely result in an expansion, thereby increasing its volume.
• On the other hand, raising the temperature of a gas while maintaining constant pressure will also cause the gas to expand.

In both cases, the amount of expansion depends on the initial conditions and how significantly the pressure or temperature changes. The overall goal is to calculate or understand the ratio of the final volume to the initial volume (\(\frac{V_2}{V_1}\)). This ratio helps determine which action causes a larger change in volume.

Pressure Conversion

Pressure conversion is a crucial step when working with gas laws, especially the Ideal Gas Law, because it ensures all units are consistent. In our example, the pressures are given in millimeters of mercury (mmHg), but it's often necessary to convert these to SI units such as Pascals (Pa) for ease of calculation.

• And to perform this transformation, we use the conversion: \(1 \, \text{mmHg} = 133.322 \, \text{Pa}\).
• Multiply each of the given pressures by this conversion factor to obtain the pressures in Pascals.

This conversion is essential because it allows us to apply the Ideal Gas Law formulas uniformly, making our calculations accurate and standard across different physical contexts. The step of converting units might seem tedious, but it prevents errors in further mathematical calculations.

Temperature Conversion

Temperature conversion is important when dealing with gas laws, especially because the Ideal Gas Law (\(PV = nRT\)) requires the temperature to be in Kelvin. Kelvin is the SI unit for temperature and is absolute, meaning it doesn’t go negative, which is crucial for accurate calculations.

To convert degrees Celsius to Kelvin, simply add 273.15 to the Celsius value:

• The conversion formula is: \(T(\text{K}) = T(\text{°C}) + 273.15\).
• For our exercise, converting both initial (\(10^{\circ} \text{C}\)) and final (\(40^{\circ} \text{C}\)) temperatures to Kelvin involves these simple steps.

Using Kelvin allows us to use the Ideal Gas Law correctly, providing precise results when we analyze the gas behavior under different pressure and temperature changes. Keeping temperature in the right unit helps us direct our calculations seamlessly.

Gas Laws

Gas laws are fundamental principles that describe how gases behave under different conditions of pressure, volume, and temperature. The Ideal Gas Law, represented as \(PV = nRT\), is one of the main gas laws used in calculations because it combines the relationships described by other laws like Boyle's and Charles's Laws. This particular law becomes exceptionally helpful when we consider changes under different conditions.

• Boyle’s Law: At constant temperature, the volume of a gas inversely relates to pressure: \(P_1V_1 = P_2V_2\).
• Charles’s Law: At constant pressure, the volume of a gas is directly proportional to its temperature: \(\frac{V_1}{T_1} = \frac{V_2}{T_2}\).

In our exercise, these principles are mirrored in determining which change elicits a larger volume increase: lowering pressure or raising temperature. By utilizing the concept of constant conditions—temperature in one case and pressure in another—we can strategically apply parts of the Ideal Gas Law to reach our desired conclusions.